WASHINGTON—Before the U.S. agreed to release five Taliban detainees in exchange for Sgt. Bowe Bergdahl, two groups of U.S. military personnel worked for more than year on alternative plans to secure his freedom—including one proposal to pay off militants and another to release an Afghan warlord.

Those efforts, which ran on parallel tracks, underscore the extent to which U.S. agencies pursued competing approaches to free the only...